Did you know that a hammer and chisel were used to open canned food in the past? This was before the can opener was invented. Imagine being alive at that time, having to go through that trouble just to open a can of soup. You may have noticed that most canned food has a cylindrical shape.
In this article, you will learn about the surface of a cylinder, in particular about the surface area of a cylander.
What is a Cylinder?
The term cylindrical means to have a straight parallel sides and circular cross sections.
A cylinder is a three-dimensional geometric figure with two flat circular ends and a curved side with the same cross section from one end to the other.
The flat circular ends of a cylinder are parallel to each other and they are separated or joined together by a curved surface. See the figure below.
Fig. 1. Parts of a right cylinder.
Some examples of cylindrical shapes we see every day are canned food and canned soup. The individual parts of a cylinder are shown below. The ends are circles, and if you roll out the curved surface of a cylinder you get a Rectangle!
Fig. 2. The individual part of a cylinder.
There are different types of cylinders, including:
Right circular cylinders, like in the picture above,
Half cylinders;
Oblique cylinders (a cylinder where the top is not directly above the base); and
Elliptic cylinders (where the ends are ellipses rather then circles).
In particular you will be looking at right circular cylinders here, so from now on they will just be called cylinders.
Total Surface Area of a Cylinder
Let's look at the definition of the total surface area of a cylinder.
The total surface area of a cylinder refers to the area occupied by the surfaces of the cylinder, in other words the surfaces of both circular ends and the curved sides.
The unit for the surface area of a cylinder is \( cm^2\), \( m^2\) or any other square unit.
Usually people leave off the word "total", calling it just the surface area of a cylinder. As you can see from the picture in the previous section, there are two parts to the area of a cylinder:
Let's take a look at each part.
Lateral Surface Area of a Cylinder
To make life easier, let's use some variables. Here:
Generally the area of a rectangle is just the length of the two sides multiplied together. One of those sides you are calling \(h\), but what about the other side? The remaining side of the rectangle is the one that wraps around the circle that makes up the end of the cylinder, so it needs to have a length that is the same as the circumference of the circle! That means the two sides of the rectangle are:
That gives you a lateral surface area formula of
\[ \text{Lateral surface area } = 2\pi r h.\]
Let's take a look at an example.
Find the lateral surface area of the right cylinder below.
Fig. 3. Cylinder of \(11\text{ cm}\) height and \(5\text{ cm}\) radius.
Answer:
The formula for calculating the lateral surface area is:
\[ \text{Lateral surface area } = 2\pi r h.\]
From the picture above, you know that:
\[r = 5\, \text{cm} \text{ and } h = 11\, \text{cm}.\]
Putting those into your formula gives you\[\begin{align} \mbox { Lateral surface area } & = 2 \pi r h \\& = 2 \pi \cdot 5 \cdot 11 \\& = 2 \pi \cdot 55 \\ & = 2 \cdot 3.142 \cdot 55 \\ & \approx 345.62 \text{ cm}^2 .\end{align} \]
Now on to the total surface area!
Formula for the Surface Area of a Cylinder
A cylinder has different parts which means it has different surfaces; the ends have their surfaces and the rectangle has its surface. If you want to calculate the surface area of a cylinder, you need to find the area occupied by both the rectangle and the ends.
You already have a formula for the lateral surface area:
\[ \text{Lateral surface area } = 2\pi r h.\]
The ends of the cylinder are circles, and the formula for the area of a circle is
\[ \text{Area of a circle } = \pi r^2.\]
But there are two ends to the cylinder, so the total area of the ends is given by the formula
\[ \text{Area of cylinder ends } = 2\pi r^2.\]
The surface area occupied by both the rectangle part and the ends is called the total surface area. Putting together the formulas above gives you the total surface area of a cylinder formula
\[\text{Total surface area of cylinder } = 2 \pi r h + 2\pi r^2.\]
Sometimes you will see this written as
\[\text{Total surface area of cylinder } = 2 \pi r (h +r) .\]
Calculations for the Surface Area of Cylinders
Let's take a look at a quick example that uses the formula you found in the previous section.
Find the surface area of a right cylinder whose radius is \(7 \text{ cm}\) and its height is \(9 \text{ cm}\).
Answer:
The formula for finding the surface area of a right cylinder is
\[\text{Total surface area of cylinder } = 2 \pi r (h +r) .\]
From the question you know the value of the radius and height are
\[r = 7\, \text{cm} \text{ and } h = 9\, \text{cm}.\]
Before you proceed, you should make sure that the values of the radius and height are of the same unit. If they aren't you will need to convert units so they are the same!
The next step is to substitute the values in the formula:\[ \begin{align}\mbox {Total surface area of cylinder } & = 2 \pi r (r + h) \\& = 2 \pi \cdot 7 (7 + 9) \\& = 2 \pi \cdot 7 \cdot 16 \\& = 2 \pi \cdot 112 \\& = 2 \cdot 3.142 \cdot 112. \\ \end{align}\]
Don't forget your units when writing the answer! So for this problem, the total surface area of the cylinder is \(112 \, \text{cm}^2\).
You may be asked to find an approximate answer to one decimal place. In that case, you can plug it into your calculator to get that the total surface area is approximately \(703.8 \, \text{cm}^2 \).
Let's take a look at another example.
Find the surface area of a right cylinder given the radius to be \(5\, \text{ft}\) and the height to be \(15\, \text{in}\).
Answer:
The formula for finding the surface area of a right cylinder is:
\[\text{Total surface area of cylinder } = 2 \pi r (h +r) .\]
From the question you know the values of the radius and height are:
\[r = 5\, \text{ft} \text{ and } h = 15\, \text{in}\]
Stop! These are not the same units. You need to convert one to the other. Unless the question states what units the answer should be in, you can pick either one to convert. In this case it isn't specified, so let's convert the radius to inches. Then
\[ 5 \, \text{ft} = 5 \, \text{ft} \cdot \frac{ 12\, \text{in}}{1 \, \text{ft}} = 60 \, \text{in}.\]
Now you can substitute the values
\[r = 60\, \text{in} \text{ and } h = 15\, \text{in}\]
in the formula to get
\[\begin{align} \mbox {Total surface area of cylinder }& = 2 \pi r (r + h) \\& = 2 \pi \cdot 60 (60 + 15) \\& = 2 \pi \cdot 60 \cdot 75 \\ & = 2 \pi \cdot 4500 \\& = 9000 \pi \text{in}^2. \end{align} \]
What happens if you cut a cylinder in half?
Surface Area of a Half Cylinder
You have learned about the surface area of a cylinder, but let's see what happens when the cylinder is cut in half lengthwise.
A half cylinder is obtained when a cylinder is cut longitudinally into two equal parallel parts.
The figure below shows what a half-cylinder looks like.
Fig. 4. A Half Cylinder.
When you hear the word 'half' in mathematics, you think about something divided by two. So, finding the surface area and the total surface area of a half cylinder involves dividing the formulas for a right cylinder (a complete cylinder) by two. That gives you
\[\text{Surface area of half cylinder } = \pi r (h +r) .\]
Let's take a look at an example.
Calculate the surface area of the half cylinder below. Use the approximation \(\pi \approx 3.142\).
Fig. 5. Half cylinder.
Answer:
From the figure above, you have
\[r= 4\, \text{cm}\text{ and } h= 6\, \text{cm}. \]
The formula you would use here is:
\[\text{Surface area of half cylinder } = \pi r (h +r) .\]
Substituting values into the formula,
\[ \begin{align} \mbox {Surface area of half cylinder } & = 3.142 \cdot 4 \cdot (6+4) \\ &= 3.142 \cdot 4 \cdot 10 \\& = 75.408\, \text{cm}^2 \end{align} \]
Surface Area of a Capped Half Cylinder
With the surface area of a capped half cylinder, it is more than just dividing by two. There is something else you have to consider. Remember the cylinder you are dealing with is not complete, in other words it certainly wouldn't hold water! You can cap it by adding a rectangular section over the cut part. Let's take a look at a picture.
Fig. 6. Showing the rectangle surface of a half cylinder.
You just need the area of that rectangle surface you capped the cylinder with. You can see it has the same height as the actual cylinder, so you just need the other side. It turns out that is the diameter of the circle, which is the same as twice the radius! So
\[ \begin{align} \text{Surface area of capped half cylinder } &= \text{Surface area of half cylinder } \\ &\quad + \text{Area of rectangle cap} \\ &= \pi r (h +r) + 2rh.\end{align}\]
Let's take a look at an example.
Find the surface area of the capped half cylinder in the picture below.
Fig. 7. Half cylinder.
Solution.
The formula you will use here is
\[\text{Surface area of capped half cylinder } = \pi r (h +r) + 2rh.\]
The figure above shows the value of the diameter and the height:
\[\mbox { diameter } = 7\, \text{cm} \text{ and } h = 6\, \text{cm}. \]
But the formula calls for the radius, so you need to divide the diameter by \(2\) to get
\[ r= \frac{7} {2} \, \text{cm}. \]
So, the values you need are
\[ r = 3.5\, \text{cm} \text{ and } h= 6\, \text{cm}. \]
So, the surface area will be:
\[ \begin{align} \text{Surface area of half capped cylinder } &= \pi r (h +r) + 2rh \\ &= \pi\left(\frac{7}{2}\right)\left( \frac{7}{2} +6\right) + 2\left(\frac{7}{2}\right) 6 \\ &= \pi \left(\frac{7}{2}\right) \left(\frac{19}{2}\right) + 42 \\ &= \frac{133}{4}\pi + 42 \, \text{cm}^2. \end{align} \]
If you are asked to give an approximate answer to two decimal places, you would find that the surface area of the capped half cylinder is approximately \(146.45\, \text{cm}^2\).
Surface Area of Cylinder - Key takeaways
- The term cylindrical means to have a straight parallel sides and circular cross sections.
- The surface area of a cylinder refers to the area or space occupied by the surfaces of the cylinder i.e the surfaces of both bases and the curved sides.
- The formula for calculating the lateral surface area of a right cylinder is \(2 \pi r h\).
- The formula for calculating the surface area of a right cylinder is \(2 \pi r (r + h) \).
- The formula for calculating the surface area of a half cylinder is \(\pi r (h +r) \).
- The formula for calculating the surface area of a capped half cylinder is \( \pi r (h +r) + 2rh \).
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